Related papers: Unexpected Delta-Function Term in the Radial Schro…

The notion of a delta shock wave and a singular shock wave was introduced and employed by different authors, and it was shown that a large class of Riemann problems can be solved globally with these additional building blocks. The aim of…

In this paper we derive an expression for the dynamic electric polarizability of a particle bound by a double delta potential for frequencies below and above the absolute value of the particle's ground state energy. The derived expression…

We argue that the "reduced wave function", proposed recently [Phys.Rev.Lett. {\bf 75}, 2255 (1995)], contains conditional and restricted information on the reduced system. The concept of "reduced wave function" can thus not represent a…

We consider the nonlinear Schr\"odinger equation \begin{equation*} \Delta u = \big( 1 +\varepsilon V_1(|y|)\big)u - |u|^{p-1}u \quad \text{in} \quad \mathbb{R}^N, \quad N\ge 3, \quad p \in \left(1, \frac{N+2}{N-2}\right).\end{equation*} The…

Mathematical justifications are given for several integral and series representations of the Dirac delta function which appear in the physics literature. These include integrals of products of Airy functions, and of Coulomb wave functions;…

In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schr\"odinger equation, which is solved for the wave function, bound…

Generalized eigenfunctions of the 3-dimensional relativistic Schr\"odinger operator $\sqrt{\Delta} + V(x)$ with $|V(x)|\le C < x >^{{-\sigma}}$, $\sigma > 1$, are considered. We show that the generalized eigenfunctions can be expressed as…

In two and three dimensions, the standard treatment of the scattering problem for a multi-delta-function potential, $v(\mathbf{r})=\sum_{n=1}^N\mathfrak{z}_n\delta(\mathbf{r}-\mathbf{a}_n)$, leads to divergent terms. Regularization of these…

Schrodinger equation with two-component wave function which describes a relativistic spin 1/2 particle in a weak electromagnetic field is obtained. In the same approximation Schrodinger equation with traditional norm condition and…

The theory of interaction at one point is developed for the one-dimensional Schrodinger equation. In analog with the three-dimensional case, the resulting interaction is referred to as the Fermi pseudo-potential. The dominant feature of…

For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent)…

The nonlinear Schroedinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schroedinger equation can be solved analytically in…

There is a fundamental difference between the classical expression for the retarded electromagnetic potential and the corresponding retarded solution of the wave equation that governs the electromagnetic field. While the boundary…

We present a conditionally exactly solvable singular potential for the one-dimensional Schr\"odinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general…

We consider the steady fractional Schr\"odinger equation $L u + V u = f$ posed on a bounded domain $\Omega$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-\Delta)^s$; $V\ge 0$ is a potential…

The functional Schr\"odinger representation of a scalar field on an $n$-dimensional static space-time background is argued to be a singular limiting case of the hypercomplex quantum theory of the same system obtained by the precanonical…

We study the effect of a $\delta$ distribution potential placed at $x_0\geq 0$ and multiplied by a parameter $\alpha$ on a quantum mechanical particle in an infinite square well over the segment $\left[-\,\frac{L}{2},\frac{L}{2}\right]$. We…

Whenever the Breit-Wigner amplitude appears in a calculation,there are many instances (e.g., Fermi's two-level system and the Weisskopf-Wigner approximation) where energy integrations are extended from the scattering spectrum of the…

The one-particle three-dimensional Dirac equation with spherical symmetry is solved for the Hulthen potential. The s-wave relativistic energy spectrum and two-component spinor wavefunctions are obtained analytically. Conforming to the…

The behaviour of the spectral edges (embedded eigenvalues and resonances) is discussed at the two ends of the continuous spectrum of non-local discrete Schr\"odinger operators with a $\delta$-potential. These operators arise by replacing…